On the critical Choquard equation with potential well

TL;DR

This paper studies the existence and behavior of solutions to a nonlinear Choquard equation with potential well, demonstrating solution localization for large parameters and multiple solutions via topological methods.

Contribution

It establishes the existence of ground state solutions localized near the potential well and characterizes their asymptotic behavior as the parameter grows, introducing new analytical techniques.

Findings

Existence of ground state solutions localized near the potential well for large λ.

Asymptotic behavior of solutions as λ approaches infinity.

Multiple solutions exist for certain parameter ranges using Lusternik-Schnirelmann theory.

Abstract

In this paper we are interested in the following nonlinear Choquard equation $$ -\Delta u+(\lambda V(x)-\beta)u =\big(|x|^{-\mu}\ast |u|^{2_{\mu}^{\ast}}\big)|u|^{2_{\mu}^{\ast}-2}u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $\lambda,\beta\in\mathbb{R}^+$, $0<\mu<N$, $N\geq4$, $2_{\mu}^{\ast}=(2N-\mu)/(N-2)$ is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality and the nonnegative potential function $V\in \mathcal{C}(\mathbb{R}^N,\mathbb{R})$ such that $\Omega :=\mbox{int} V^{-1}(0)$ is a nonempty bounded set with smooth boundary. If $\beta>0$ is a constant such that the operator $-\Delta +\lambda V(x)-\beta$ is non-degenerate, we prove the existence of ground state solutions which localize near the potential well int $V^{-1}(0)$ for $\lambda$ large enough and also characterize the asymptotic behavior of the solutions as the parameter $\lambda$ goes to infinity. Furthermore, for any $0<\beta<\beta_{1}$, we are able to find the existence of multiple solutions by the Lusternik-Schnirelmann category theory, where $\beta_{1}$ is the first eigenvalue of $-\Delta$ on $\Omega$ with Dirichlet boundary condition.